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There's something I don't get regarding the orbits of photons in Schwarzschild geometry.

As well known, by solving geodesics equation for null rays, you get that photons can be in a (unstable) circular orbit at r=3 M. However, if you look at the causal diagram for Schwarschild geometry (in Schwarzschild coordinates, say), then you see that the light cones are still open.

So, according to this diagram, a photon should go left (towards decreasing r) or right (increasing r), but not straight at r=cst!! Is the solution somewhere hidden in the two other coordinates theta and phi?

I have another, but related question. The light cone gets degenerate with no extension at r=2M, as you see again on this causal diagram. In other words, the horizon is a null surface, which is a general property anyway. But if the horizon is a null surface, why don't photons actually move along this null surface? Instead, they just cross inwards.

To summarize, I would have expected the photon sphere to be precisely the horizon, since for me I have this (wrong) idea in head : null surface <-> photon sphere

Where is the mistake?

Thanks a lot